Combinatorial positivity and monotonicity of discrete intrinsic volumes

Let $P\in\mathcal P(\mathbb{Z}^d)$ have dimension $r$, and let the $k$-th discrete intrinsic volume be represented in the binomial basis by

$$V_k(tP)=h^{V_k}_0(P)\binom{t+r}{r}+h^{V_k}_1(P)\binom{t+r-1}{r}+\cdots+h^{V_k}_r(P)\binom{t}{r}.$$

A valuation is combinatorially positive if all its binomial-basis coefficients are nonnegative, and combinatorially monotone if these coefficients do not decrease under inclusion $P'\subseteq P$. The discrete intrinsic volume conjecture. For any $k\in\{0,1,\dots,d\}$, the $k$-th discrete intrinsic volume is combinatorially positive and monotone. This conjecture asks whether the positivity and monotonicity already known for the discrete volume and solid angle valuations also hold for all discrete intrinsic volumes.