Filter on client cert EKU

Is it possible to configure my SSL client profile so it will only request certificates with a particular EKU from Internet Explorer (in my case)?

Background: when I currently try to authenticate to the F5 with my smart card based credential, windows (IE specifically) prompts me with its cert dialog box. The problem is that it shows me two of my certificates. Since they have the same CN, they appear identical in the dialog box. Only one of them is my authentication cert and will work. The other one, while present and available for me to pick, will cause the cert auth to fail.

I'd like to get rid of this second cert as a selectable option.

Any ideas?