Gem-complexity bound for closed orientable 3-manifolds

Let $M$ be a closed orientable 3-manifold, let $c(M)$ denote its complexity, and let $k(M)$ denote its gem-complexity.

Gem-complexity bound. One has

$$k(M) \le 5+2c(M).$$

This conjecture was stated in connection with possible relations between Matveev complexity and gem-complexity; the supplied text gives no resolution status.