Jockusch's edge lower-bound conjecture for cubical polytopes

Let $K$ be a cubical polytope of dimension $d\geq 3$, with $V$ vertices and $E$ edges. Jockusch's conjecture.

$$E\geq \frac{d+1}{2}V-2^{d-1}.$$

The source attributes this conjecture to Jockusch and discusses partial results toward it.