Fu's Weyl principle for Holmes–Thompson intrinsic volumes

Let $V_j$ be finite-dimensional real vector spaces equipped with smooth norms $F_j$, and let $e_j\colon M\to V_j$, $j=1,2$, be smooth immersions of a smooth manifold $M$. For a valuation, its pullback along an immersion is defined by $e^*\mu(A)=\mu(e(A))$ for compact submanifolds with corners $A\subset M$. Fu's Weyl principle. If

$$e_1^*F_1=e_2^*F_2,$$

then

$$e_1^*\mu_k^{F_1}=e_2^*\mu_k^{F_2}.$$

This conjecture asserts that the Holmes–Thompson intrinsic volumes depend only on the induced Finsler metric, not on the particular immersion into a normed space. The source presents it as an open conjecture; the supplied material gives no resolution.