Equitable 2-colourability conjecture for even cycle decompositions

Let $v\geq \ell\geq 4$ be even integers, and let $I$ be a 1-factor of $K_v$. An equitably 2-colourable $\ell$-cycle decomposition of $K_v-I$ is an $\ell$-cycle decomposition whose vertices admit a red-blue colouring with colour classes differing in size by at most one on each cycle. The divisibility condition is that $\ell$ divides the number of edges of $K_v-I$:

$$\ell\mid \frac{v(v-2)}{2}.$$

Even-cycle equitable-colourability conjecture. There exists an equitably 2-colourable $\ell$-cycle decomposition of $K_v-I$ if and only if

$$\ell\mid \frac{v(v-2)}{2}.$$

The paper leaves the more general existence question for equitably 2-colourable odd cycle decompositions of the cocktail party graph open; the stated even-order conjecture is likewise presented as unresolved.