Those constants don't exist for any $d\geq4$, here is an idea of why.

For each $\varepsilon>0$ let $A_\varepsilon=\{(x_1,\dots,x_d)\in\mathbb{R}^d;\lvert (d-1)x_d-\sum_{i=1}^{d-1} x_i\rvert<\varepsilon\text{ and }\lvert\frac{(d-1)(d-2)}{2}x_d-\sum_{i=1}^{d-1}ix_i\rvert<\varepsilon\}$. Note that there is a big constant $K$ independent of $\varepsilon$ such that for all $i,j$, the intersection of $A_\varepsilon$ with span$(e_i,e_j)$ is contained in the ball $B(0,K\varepsilon)$.

Also note that $v=(1,1,\dots,1)\in A_\varepsilon$, and let $B_\varepsilon=\{x\in A_\varepsilon;d(x,span(v))\leq K\varepsilon\}$.

Then both $A_\varepsilon$ and $B_\varepsilon$ are convex and inscribed in the cube, and their intersection with span$(e_i,e_j)$ is the same $\forall i,j$, but when $\varepsilon\to0$, the volume of $A_\varepsilon$ is, up to some constant, proportional to $\varepsilon^2$ and the volume of $B_\varepsilon$ is proportional up to some constant to $\varepsilon^{d-1}$.

Those constants don't exist for any $d\geq4$, here is an idea of why.

For each $\varepsilon>0$ let $A_\varepsilon=\{(x_1,\dots,x_d)\in[-1,1]^d;\lvert (d-1)x_d-\sum_{i=1}^{d-1} x_i\rvert\leq\varepsilon\text{ and }\lvert\frac{(d-1)(d-2)}{2}x_d-\sum_{i=1}^{d-1}ix_i\rvert\leq\varepsilon\}$. Note that there is a big constant $K$ independent of $\varepsilon$ such that for all $i,j$, the intersection of $A_\varepsilon$ with span$(e_i,e_j)$ is contained in the ball $B(0,K\varepsilon)$.

Also note that $v=(1,1,\dots,1)\in A_\varepsilon$, and let $B_\varepsilon=\{x\in A_\varepsilon;d(x,span(v))\leq K\varepsilon\}$.

Then both $A_\varepsilon$ and $B_\varepsilon$ are convex and inscribed in the cube, and their intersection with span$(e_i,e_j)$ is the same $\forall i,j$, but when $\varepsilon\to0$, the volume of $A_\varepsilon$ is, up to some constant, proportional to $\varepsilon^2$ and the volume of $B_\varepsilon$ is proportional up to some constant to $\varepsilon^{d-1}$.