Conjecture on the asymptotic classes and equality of NFS minimizer series

Let $a,b,d$ be the minimizers of Problem~, and let $\mathbf A,\mathbf B\in\mathbb R[[X,Y]]$ be the series associated to $a$ and $b$, respectively. Let $\mathcal C^{[1/3,2/3]}$ and $\mathcal C^{[1/3,-1/3]}$ denote the classes of functions used in the asymptotic analysis. Asymptotic-class and series-equality conjecture. The minimizers $a,b,d$ belong respectively to the classes of functions $\mathcal C^{[1/3,2/3]}$, $\mathcal C^{[1/3,2/3]}$, and $\mathcal C^{[1/3,-1/3]}$. Moreover, the associated series satisfy

$$\mathbf A=\mathbf B.$$

The conjecture formalizes the patterns observed in the experimental computations: the minimizers have the predicted asymptotic forms, and the two series obtained from $a$ and $b$ coincide. Its resolution would justify the recurring structural assumptions used in proving existence and minimality and clarify the asymptotic expansion of the number field sieve complexity.