Hi.

The short answer is YES.
The long one;
in your preferences (Shift Ctrl P)
Interface - Windows;
Click the "Save and restore Dialog Status" radio button.

To have your layer list fully expanded on re-opening you have to "un-dock" the layer dialog.
(Click and drag the top of the dialog to undock.)

Good Luck
RGDS
Ragnar

Starting with version 0.91, coming out in stable release soon, if you close Inkscape with the Layers dialog open, it will be open the next time you open Inkscape (as long as you have settings configured properly). You can get it now as 0.91pre4 https://inkscape.org/en/download/.