# Projection of a polytope along 4 orthogonal axes

Consider the following problem:

Given an $$\mathcal{H}$$-polytope $$P$$ in $$\mathbb{R}^d$$ and $$4$$ orthogonal vectors $$v_1, …, v_4 \in \mathbb{R}^d$$, compute the projection of $$P$$ to the subspace generated by $$v_1, …, v_4$$ (and ouput it as an $$\mathcal{H}$$-polytope).

I know that the problem of computing projections along $$k$$ orthogonal vectors in NP-hard (if $$k$$ and $$d$$ are part of the input), as shown in this paper. But does it help if $$k$$ is a constant? Specifically, does it help if $$k \leq 4$$? Do we have a polynomial algorithm in this case?