This answer isn't going to fully clear this up (at least, not yet), but maybe it can help trace where that claim could have come from. (And maybe be a first stepping stone for a full answer, proving or disproving the claim.)
Let's try some OSINT. (Or, if you will, "bibliographical research".)
The Vikipedio article H-sistemo that you are referring to, currently (as of 2021-10-30; permalink) isn't well-sourced. It makes notable claims without indicating what the source or sources for said information is. That using the apostrophe in the indicated way for the H-System is a Zamenhofian recommendation is one of those unsourced claims. (Maybe this should be brought up on the article's talk page.)
Possible source: English Wikipedia ← Morsa kodo kaj Esperanto (2008) by Lenio Marobin ← ?
However the same claim or at least a similar one can be found in the English language Wikipedia, too:
Zamenhof had suggested replacing the circumflex letters with digraphs in h, the so-called "h-system", thus: ch, gh, jh, sh for ĉ, ĝ, ĥ, ĵ, ŝ; kh for ĥ and u for ŭ, with an apostrophe or hyphen to disambiguate actual sequences of these letters (e.g. ses-hora).
—Wikipedia article Esperanto orthography
The original method of working around the diacritics was developed by the creator of Esperanto himself, L. L. Zamenhof. He recommended using u in place of ŭ, and digraphs with h for other the circumflex letters. For example, ŝ is replaced by sh, as in shanco for ŝanco (chance). Where proper orthography has sh, the letters should be separated with an apostrophe or a hyphen, as in ses-hora (six-hour) or flug'haveno (airport).
—Wikipedia article Substitutions of the Esperanto alphabet
These two claims are both referenced to the same document, an article in the ILERA bulletin:
Indeed that article (pages 3-4 in the PDF) by Lenio Marobin (call sign
PY3DF), at the very end of page 4 (and of said article), also makes a similar claim:
La francaj rondanoj (precipe per paket-ra-
dio) nur uzadas la zamenhofan sistemon (ch, gh,
jh, sh, u) senprobleme, sed kiel Z. mem proponis:
se la h devas esti elparolata aparte de la c, g, j, s,
ĝi ricevas antaŭ si apostrofon ( ' ) aŭ streketon:
ses-hora, dek-harmona, senc-hava. Li ankaŭ pro-
ponis kh por ĥ.
The bulletin article itself doesn't cite any sources for this claim.
According to the article's introduction (and the addition "rekolekto de
artikoloj iam aperintaj" in its title), it seems to be an edited reprint of a previous publication:
Kompilis: Lenio Marobin,PY3DF
Post la ricevo de la artikolo de Martin,
IK2RMZ, mi konstatis, ke en mia paperteko tro-
viĝis kelkaj flavaj fotokopioj de preskaŭ nelege-
blaj originaloj. Tio incitis min re-redakti ilin, kaj
korekti kelkajn informojn laŭ la sugestoj de nia
forpasinta Henriko,F9ED. La subaj estas remaĉo
de originalo aperinta unue en la rondletero de
GEPRAF (novembre 1992), kaj poste en Eventoj
n-ro 106 (julie 1996), fare de László Szilvási,
HA-9057, ILERA n-ro 98. La Enkondukon mi
So the alleged previous publications are:
- rondletero de GEPRAF (November 1992)
- Eventoj n-ro 106 (July 1996)
presumably both by László Szilvási, call sign
HA-9057, ILERA n-ro 98.
(Or maybe that was meant to indicate that there was a third previous publication in ILERA Bulteno n-o 98? But wouldn't No. 98 have appeared after No. 70?)
At least in the retpoŝta versio (e-mail version) of Eventoj n-ro 106 available online, in the article "Telegrafiaj morsaj kontaktoj" by Henri F9ED (who doesn't seem to be the same as aforementioned László Szilvási, but the latter is listed as "respondeca eldonanto kaj ĉefredaktoro") does not contain such a claim.
Thus it might be an addition by the author of the article in ILERA Bulteno n-o 70, Lenio Marobin. So if he's still around, maybe one could ask him what his source for that claim was. (If he can still remember over 13 years later.)
What edit introduced the claim in the Esperanto Vikipedio?
That would be 54262, anonymously made through
18.104.22.168 on 2004-03-22 and without an edit summary that might give insight into sources or intentions. From that IP address's talk page I conclude (without much certainty) that it could have been Helmut Welger. According to his Vikipedio user page (permalink, Helmut Welger passed away in 2008, so we can't ask him anymore.