rigid collapse to $\aleph_1$

Suppose $\kappa$ is inaccessible (or more). Does there exist a $\kappa$-c.c. partial order $\mathbb P \subseteq V_\kappa$ that forces $\kappa = \aleph_1$, with the following property?-- Whenever $G \subseteq \mathbb P$ is generic over $V$, there does not exist any $H \not= G$ in $V[G]$ which is also $\mathbb P$-generic over $V$.

• I imagine that some Easton iteration which collapses the regular cardinals below $\kappa$ while coding the generic into some definable real might do the trick? – Asaf Karagila Sep 21 '17 at 16:12
• If $\kappa$ is weakly compact, then every real added by $\mathbb P$ exists in an intermediate model by a forcing of size $<\kappa$. – Monroe Eskew Sep 21 '17 at 16:14
• Well. Iterate the whole thing with coding the generic into a real or something? – Asaf Karagila Sep 21 '17 at 16:14
• Sounds promising. I don't know this technique. – Monroe Eskew Sep 21 '17 at 16:17
• Well. I'll definitely drop by the KGRC sometime next week. – Asaf Karagila Sep 21 '17 at 16:37

The answer is positive when $\kappa$ is Mahlo. Please email me for a draft if you are interested.
{"ft_lang_label":"__label__en","ft_lang_prob":0.8461588,"math_prob":0.99246514,"size":1446,"snap":"2019-26-2019-30","text_gpt3_token_len":431,"char_repetition_ratio":0.16851595,"word_repetition_ratio":0.11016949,"special_character_ratio":0.30912864,"punctuation_ratio":0.10486891,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9963759,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2019-06-20T04:22:18Z\",\"WARC-Record-ID\":\"<urn:uuid:ac4b15b8-9429-4155-9fee-f5043e675ee0>\",\"Content-Length\":\"121395\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:f0ef9760-1103-49cf-8f2b-9ef063ac994e>\",\"WARC-Concurrent-To\":\"<urn:uuid:a8d6fb6d-29d0-463a-a0e7-9b47545d3fff>\",\"WARC-IP-Address\":\"151.101.129.69\",\"WARC-Target-URI\":\"https://mathoverflow.net/questions/281684/rigid-collapse-to-aleph-1\",\"WARC-Payload-Digest\":\"sha1:TZEHUZQL2TGXGIIQYRBF7GPJUDPB3YDT\",\"WARC-Block-Digest\":\"sha1:2GNKCP56L3E44UD4XILAPBFXQGKJ7NPA\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2019/CC-MAIN-2019-26/CC-MAIN-2019-26_segments_1560627999130.98_warc_CC-MAIN-20190620024754-20190620050754-00498.warc.gz\"}"}