Uniform distribution conjecture for the generalized coupon collector time

Let $n\geq 1$, let $c\in\{1,\ldots,n\}$, and let $p=(p_1,\ldots,p_n)\in(0,1)^n$ satisfy $p_1+\cdots+p_n\leq 1$. Write

$$p_0=1-(p_1+\cdots+p_n),\qquad v=(v_1,\ldots,v_n),\quad v_i=\frac{1-p_0}{n},\qquad u=\left(\frac1n,\ldots,\frac1n\right).$$

Here $T_{c,n}(p)$ denotes the time needed in the generalized coupon collector problem with coupon distribution $p$. Uniform distribution conjecture. For every $k\geq 0$,

$$\mathbbm{P}\{T_{c,n}(p)>k\}\geq\mathbbm{P}\{T_{c,n}(v)>k\}\geq\mathbbm{P}\{T_{c,n}(u)>k\}.$$

Equivalently, the complementary distribution function of $T_{c,n}$ is minimized by the uniform distribution, with the intermediate equalization distribution $v$ providing the stated comparison. The paper presents this as a new conjecture; no resolution is supplied in the given text.