Setting up subsonic behind an HTTPS reverse proxy

I have a mostly working setup with nginx as a reverse proxy with TLS termination, using Let's Encrypt certificates. Most things seem to be working fine, except for a few pages that don't load, specifically, the settings page, and the “Playing” entry in the top menu.
The reason those don't work is that they return redirects to plain-text HTTP resources on the same domain, which are rejected by the browser, since the origin is loaded over HTTPS. (And as I'm typing this, I noticed that other redirects also lead to HTTP resources, like the login page; the login page in particular doesn't break everything, since it's loaded as a top-level browser context, so the browser just follows the redirect, which then gets redirected back to HTTPS by nginx anyway.)
I have tried to change the “Access your server over the Internet using an easy-to-remember address.” setting to use my custom HTTPS root URL, but it doesn't seem to make any difference – that would probably have something to do with the message that gets displayed below:
I have no idea why Subsonic throws that exception – I have both ISRG Root X1, and DST Root CA X3 in my system CA bundle, and no other client appears to have any trouble verifying the reverse proxy. Could it be that the HTTPS client library used by Subsonic doesn't support TLS1.2?
Either way, any idea if there's anything I can do here to make Subsonic aware that it's being served to clients over HTTPS?
The reason those don't work is that they return redirects to plain-text HTTP resources on the same domain, which are rejected by the browser, since the origin is loaded over HTTPS. (And as I'm typing this, I noticed that other redirects also lead to HTTP resources, like the login page; the login page in particular doesn't break everything, since it's loaded as a top-level browser context, so the browser just follows the redirect, which then gets redirected back to HTTPS by nginx anyway.)
I have tried to change the “Access your server over the Internet using an easy-to-remember address.” setting to use my custom HTTPS root URL, but it doesn't seem to make any difference – that would probably have something to do with the message that gets displayed below:
Status: Could not connect to https://my.subsonic.domain. (SSLException)
I have no idea why Subsonic throws that exception – I have both ISRG Root X1, and DST Root CA X3 in my system CA bundle, and no other client appears to have any trouble verifying the reverse proxy. Could it be that the HTTPS client library used by Subsonic doesn't support TLS1.2?
Either way, any idea if there's anything I can do here to make Subsonic aware that it's being served to clients over HTTPS?