I tried to formalize the idea from the post k<457a_0 based on “roots” to catch more cycles (from the 1-cycle side and from the high cycle side) but it seems I overlooked the fact that some “roots” can go as low as 2a_0/3.
I couldn’t do better then k<304a_0 (sorry, I used the notations of my previous paper)
From that we can say that integer cycles starting with at least j>\log_2(\frac{x}{304}+1) consecutive “1” in the parity vector of its a_0 can use a_0\geqslant 2^j-1>\frac{x}{304} to show that k=\lceil x \log_23\rceil (1-cycle side), and x<a_{max}\leqslant304a_0 also have k=\lceil x \log_23\rceil (high cycle side), something like using a_{max}<(\frac{3^x}{2^x})a_0<304a_0 so that consecutive “climbs” can be more than 2