Gem-complexity formula for lens spaces

Let $L(p,q)$ be a lens space, and let $k(L(p,q))$ denote its gem-complexity. Let $S(p,q)$ be the quantity used in the paper's lens-space complexity formulas.

Lens-space gem-complexity conjecture. For every $p\ge 2$,

$$k(L(p,q))=2\cdot S(p,q)-1.$$

The conjecture is motivated by the preceding proposition, which gives this formula for several classes of lens spaces; the supplied text gives no resolution status.