Zagier's conjecture on the 2-adic valuations of Laurent coefficients

Let $m=2^n m_0$, where $m_0$ is odd, and let $u= u_2$ denote the 2-adic valuation. Define $u((k)!)$ by the 2-adic valuation of the factorial. Zagier's conjecture. For the Laurent coefficients $b_m$ of the exterior Riemann mapping function, the following identities hold: if $n=0$, then $- u(b_m)= u((2m+2)!)$; if $n=1$, then

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

where $\epsilon(m_0)=0$ if $m_0\equiv11\pmod{12}$ and $\epsilon(m_0)=1$ otherwise; and if $n=2$, then

$$- u(b_m)= u(((2m-25)/7)!)+\epsilon(m_0),$$

where $\epsilon(m_0)$ has period $28$. These formulas refine the known upper bound $- u(b_m)\leq u((2m+2)!)$, with equality for odd $m$, and describe the observed periodic structure of the denominator exponents; the stated patterns for $n=1$ and $n=2$ remain conjectural.