Let $k \geq 2$ be an integer. Put $[k] = \{1, \cdots, k\}$. Let $\mathcal{P} = \{p_1, \cdots, p_k\}$ be a set of $k$ primes. For every subset $S \subseteq [k]$ put $d_S = \prod_{j \in S} p_j$. The empty product is equal to $1$, by convention.

For every $k \geq 2$, does there always exist a set of primes $\mathcal{P}$ of cardinality equal to $k$ such that for every partition $A \sqcup B = [k]$ we have $d_A + d_B = 2^k q$, where $q$ is an odd prime?