Internet Explorer does not display a certificate selection dialog when entering a site with authorization using them - how to treat it?

Question

Internet Explorer

There is a computer with Windows 10, with a certificate installed on it from some closed platform, where access is only from IE (probably because of addons like rutoken addin for accessing the CIPF). The certificate is visible in the system, the media with the key - too. When entering the site, instead of choosing a certificate, a TLS error immediately pops up, which appears if an incorrect certificate is provided to the site (or not provided, which is the same for them). Kaspersky is installed on the computer, they tried to disable it, zero effect. The certificate itself is validated locally. Where to dig?

Answer 1

1) Has the site been added to IE Trusted Sites?
2) ActiveX activated?
3) Is there a signature plugin installed (CryptoPRO plugin, Lanit, or whatever should be used there)?
4) Have you tried reinstalling the crypto provider (this is at worst)))?