According to J. Walker, Ronald Sprague has proved that this trajectory does not contain a palindrome. [I would like a reference for this.] Another proof has been given by Klaus Brockhaus.
10110 is the smallest number with this property in base 2. The analogous number in base 10 is believed to be 196, but its trajectory (see A006960) has never been proved not to contain a palindrome.
The binary numbers have a regular pattern with cycle length 4:
a(4k) = 101^(k+1)010^(k+1) for k >= 1,
a(4k+1) = 1101^(k1)0001^(k1)01 for k >= 2,
a(4k+2) = 101^(k+1)010^(k+2) for k >= 0,
a(4k+3) = 110^(k+1)101^(k)01 for k >= 1, where ^ stands for repeated concatenation.  A.H.M. Smeets, Feb 03 2019
From A.H.M. Smeets, Feb 11 2019: (Start)
Pattern with cycle length 4 represented by contextfree grammars with production rules:
S_a > 10 T_a 00, T_a > 1 T_a 0  1101;
S_b > 11 T_b 01, T_b > 0 T_b 1  1000;
S_c > 10 T_c 000, T_c > 1 T_c 0  1101;
S_d > 11 T_d 101, T_d > 0 T_d 1  0010;
see also A058042 for similar grammars for the binary represented trajectory of 77. (End)
