The polynomial bound conjecture for real cyclotomic class numbers

Let $m$ be a positive integer, let $\mathbb{Q}(\zeta_m+\zeta_m^{-1})$ be the maximal real subfield of the $m$th cyclotomic field, and let $h^+(m)$ denote its class number. Polynomial bound conjecture. There is a fixed polynomial $\operatorname{poly}$ such that, for all integers $m$,

$$h^+(m)\leq \operatorname{poly}(m).$$

This is presented as a precise form of the folklore conjecture that the real cyclotomic class number is small but hard to compute. Its resolution would support the claimed efficiency benefits for the cyclotomic-field quantum algorithms, but no resolution is supplied in the source.