Lockdown is good for developing mathematical websites, I guess. I've pushed another update to grassmannian.info: it now shows some basic information about the derived categories of generalised Grassmannians, namely when it is known that a full exceptional collection exists. An important folklore conjecture states that this is the case, but as you can see when you enable the coloring in the table, we are still a long way off of settling it.

What I never realised is that, out of all known cases, only one is not (co)minuscule or (co)adjoint. These are somehow the easiest varieties, and for the majority of them (see below) a full exceptional collection is known. The unique example outside this class is constructed by Guseva in On the derived category of $\mathrm{IGr}(3,8)$ (denoted $\mathrm{SGr}(3,8)$ on grassmannian.info.

Out of the (co)minuscule or (co)adjoint varieties, the ones for which an exceptional collection is not yet constructed are:

If you manage to construct an exceptional collection in one of these cases, I will give you a large bar of your preferred Belgian chocolate, if you need some further incentive to think about it.

I've also completed copying the information from Bourbaki's appendix.