The lifting conjecture for Type-2 isomorphic circulant graphs

Let $C_{n_1}(R_1)$ and $C_{n_2}(R_2)$ be circulant graphs satisfying

$$C_{n_1}(R_1)\Box C_{n_2}(R_2)=C_{n_1n_2}(R)\cong C_{n_1n_2}(n_2R_1\cup n_1R_2)$$

for some $R$ and $\gcd(n_1,n_2)=1$. Suppose that, for some $m_1$, $t_1$, and $S_1$,

$$C_{n_1}(R_1)\cong_{T2_{n_1,m_1,t_1}}C_{n_1}(S_1).$$

Lifting conjecture. Then, for some $S$,

$$C_{n_1n_2}(R)\cong_{T2_{n_1n_2,m_1,t_1n_2}}C_{n_1n_2}(S).$$

The conjecture asserts that a Type-2 isomorphism on one relatively prime factor lifts to the corresponding product circulant graph; the source does not state whether it has been resolved.