I'd like any insight or references to the following two conjectures (see the glossary below for definitions):

Conjecture 1: For any string $x$, there exists a longest common subsequence of $x$ and its reversal $x^R$ that is a palindrome.

Conjecture 2: For any string $x$ over a two-letter alphabet, all longest common subsequences of $x$ and $x^R$ are palindromes.

Conjecture 2 is not true for strings over a three-letter alphabet, a counterexample being $abacbab$, which has $abcab$ and $bacba$ as longest common subsequences.

Glossary:

A **string** (or **word**) is any finite sequence of objects ("letters") drawn from some finite set (the "alphabet").

For any string $x = x_1x_2\cdots x_{n-1}x_n$ of length $n$, the **reversal** of $x$ is $x^R := x_nx_{n-1}\cdots x_2x_1$.

A string $x$ is a **palindrome** if $x = x^R$.

A string $x$ is a **subsequence** of a string $y$ if $x$ results from $y$ by removing zero or more letters (in arbitrary locations, closing up any gaps that result).

A **longest common subsequence** (**LCS**) of two strings $x$ and $y$ is a string $z$ that is a subsequence of both $x$ and $y$ such that no string longer than $z$ has this property. Generally, $x$ and $y$ may have several different LCSs. There is a well-known algorithm to find an LCS of two given strings that runs in quadratic time (see e.g., Cormen, Leiserson, Rivest, and Stein, *Introduction to Algorithms*).