Current evidence is insufficient for encoding, as it is unclear whether this is a unifiable variant of an existing encoded character. Possibly a variant of U+6896 梖?

閉鎖登記簿 (Land registration record, which was used until computerization in 2007)

旧土地台帳 (Land registration record, which was used from the Meiji era to the 1960s)

[The following comment is raised by Mr. Ma Shijie.]

Similar to 椇(U+6907).

Is there any pronunciation data to confirm whether it is a variant of either 椇 or 梖 or not?

▲ 塚田雅樹: 登記・供託オンライン申請システムに現れる地名を表すUnicode未符号化文字, 日本漢字學會報　第4号, p. 132

The new evidence still does not clarify the reading, and therefore cognateness with U+6907 椇 or U+6896 梖. The evidence in Comment #12948 is more informative. The content of footnote no. 68 in that image could be crucial.

Seems also registered in the CNS database: https://www.cns11643.gov.tw/wordView.jsp?ID=809266

cf.

01890

木 75.7.4

TC-5931

TS 11 · IDS ⿱沈木

The evidence in Comment #12948 suggests that its reading is つき (tsuki), but also that it is likely cognate with U+6896 梖. Given that this ideograph is in CNS 11643 per Comment #13878, TCA's feedback would be helpful.

Comment on #12948:
As far as I know about Kagoshima dialect, unlike most of the dialects in Japan, words can END in plosives. In the evidence, it writes the pronunciation of 髙 as タッ, which is actually resulted in the loss of some information - Kagoshima dialect pronounce it as /tak/ (FYI, standard Japanese is /taka/), different to so-called 促音 in Japanese, the /k/ sound would NEVER change according to the subsequent consonant. It is just a *coincidence* that the next consonant in this place name is also /k/.
The evidence also shows that the proposed character is pronounced as クッ, but we do not know whether it is /kuk/, /kut/, /kus/ or any other possibilities. It would be better if we could get more information about the complete pronunciation, to judge what its semantic really is.