Nonnegativity conjecture for the invariant c of integral polytopes

Let $P$ be an integral polytope, and let

$$c(P)=\sum_{F\subset P}(-1)^{\operatorname{codim}(F)}(\dim(F)+1)!\operatorname{Vol}(F),$$

where the sum runs over all nonempty faces $F$ of $P$. Nonnegativity conjecture. The invariant $c(P)$ is nonnegative:

$$c(P)\geq 0.$$

Numerical experiments and examples, including the hypersimplex $\Delta(3,6)$, support this conjecture. The claim is presented as an open conjecture for integral polytopes.