Retain API filter string

**DocFX Version Used**: 2.31.0.0

**Template used**: `default`

**Steps to Reproduce**:
1. On the API Documentation web page, enter text in the "Enter here to filter..." text box
2. Click on any of the links filtered API links

**Expected Behavior**:
When the new page is loaded, the filter text is retained in the filter box, and the API links on the left hand side remain filtered on the filter text. It would also be ideal to have a tiny button to clear the box when needed.

**Actual Behavior**:
When the new page is loaded, the filter text is cleared from the filter box, and _all_ API links on the left hand side are shown, requiring entering the same filter text again when navigating to a related API (which is very common.)