Freiman's volume conjecture with dimension

Let $k,T,d\bin \mathbb{N}$, and let $c,b$ satisfy

$$T=(d+c)k-\binom{d+c+1}{2}+d+b+1,$$

where

$$1\leq c\leq k-d-1,\qquad 0\leq b\leq k-d-c-1.$$

Here $\operatorname{vol}(k,T,d)$ is the maximum volume of a set of integers of cardinality $k$, doubling $T$, and additive dimension $d$. Freiman's volume conjecture. For these parameters,

$$\operatorname{vol}(k,T,d)=2^{c-1}(k-c+b)+1.$$

This refines a conjecture of Freiman by incorporating additive dimension and gives an explicit extremal volume in terms of the doubling parameters. The source does not provide evidence of a resolution.