Zagier-type conjecture on the 2-adic valuations of Taylor coefficients

Let $m=2^n m_0$, where $m_0$ is odd, and let $u= u_2$ denote the 2-adic valuation. Let $a_m$ be the Taylor coefficients of the reciprocal Riemann mapping function. Zagier-type conjecture. For the $a_m$ coefficients, if $n=0$, then

$$- u(a_m)= u((2m-2)!);$$

if $n=1$, then

$$- u(a_m)= u(((2m-2)/3)!)+\epsilon(m_0),$$

where $\epsilon(m_0)=1$ if $m_0\not\equiv3\pmod{12}$, while otherwise it follows the pattern in the cited table. This extends the conjectured valuation patterns for the Laurent coefficients $b_m$ to the Taylor coefficients $a_m$. The $n=0$ identity agrees with the known equality case for odd $m$; the $n=1$ pattern, including its exceptional values, remains conjectural.