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?