Exposed Form: Send only selected filters
I can confirm that on page load the option jQuery UI slider of the Better Exposed Filters module does apply the minimum value set in options. 

The minimum value is displayed in the slider, so it's a visible cue that it will be displayed. 

If you delete the number in the slider and then apply other filters, the slider filter will not be applied! 

But the minimum value will still be visible in the slider on page reload (if you're not using Ajax in Views). 

I couldn't find any similar issues so I created a new one: jQuery UI slider Exposed filter always applied

Add your test results/observations there.