Reversal-multiplication polynomiality conjecture

Let $A$ be a positive integer and let $A^{*}$ denote its reversal. A pair $(A,A^{*})$ is polynomial if the multiplication $A \times A^{*}$ can be performed without carry, and $A \times A^{*}$ is a palindrome when it equals its reversal. Reversal-multiplication conjecture. If $A \times A^{*}$ is a palindrome, then $(A,A^{*})$ is a polynomial pair.

Polynomial pairs $(A,A^{*})$ are known to produce palindromic products, and the converse is suggested by computations involving the corresponding reversal-multiplication sequence. The conjecture concerns whether every palindromic product arising from reversal multiplication is carry-free.