Vinogradov's main conjecture for the mean value $J_{r,d}(X)$

Let $J_{r,d}(X)$ denote the number of solutions in integers $1\leq x_1,\ldots,x_{2r}\leq X$ to

$$x_1^m+\cdots+x_r^m=x_{r+1}^m+\cdots+x_{2r}^m,\qquad 1\leq m\leq d,$$

where $r,d\geq 1$, and set $D=d(d+1)/2$. Vinogradov's main conjecture. For every $r\geq 1$, $d\geq 1$ and $\epsilon>0$,

$$J_{r,d}(X)\ll_{r,d,\epsilon}X^\epsilon\left(X^r+X^{2r-D}\right).$$

This is the main conjecture in Vinogradov's mean value theorem and supplies the sharp expected estimate for the number of solutions to the associated system of diagonal equations. The source notes that the bound is known trivially for $d=1,2$, for all $r$, and records results of Wooley proving it for all $r$ when $d=3$, for $r\geq d(d-1)$ when $d\geq4$, and for $100\%$ of the critical interval $1\leq r\leq D$; the remaining cases are not established in the source.