  In his work on the twenty vertex model, Di Francesco [Electron. J. Combin.
28(4) (2021), Paper No. 4.38] found a determinant formula for the number of
configurations in a specific such model, and he conjectured a closed form
product formula for the evaluation of this determinant. We prove this
conjecture here. Moreover, we actually generalize this determinant evaluation
to a one-parameter family of determinant evaluations, and we present many more
determinant evaluations of similar type - some proved, some left open as
conjectures.
