[math-fun] Palindromes multiplied

Hello Math-Fun, is this seq finite (and correct)? S = 1,2,3,11,4,22,101,5,111,6,1001,7,88,77,8,1111,9,... (my knowledges about repunits and palindromes are close to the palindrome 0 themselves) Definition: Lexicographically earliest seq of distinct palindromes such that the product of two successive palindromes is a palindrome. Best, É.