The direct-product conjecture for clean graphs over rings

Let $R_1$, $R_2$, $P_1$, and $P_2$ be rings such that

$$Cl_2(R_1)\cong Cl_2(R_2)$$

and

$$P_1\cong P_2.$$

Direct-product conjecture. The clean graphs satisfy

$$Cl_2(R_1\times P_1)\cong Cl_2(R_2\times P_2).$$

This conjecture proposes that isomorphic clean graphs are preserved when taking direct products with isomorphic rings. The supplied text gives no resolution or further evidence beyond its derivation as a generalization of the preceding theorem.