Overflow conjecture for families of bounded diameter

Let $[n]=\{1,\ldots,n\}$ and let $2^{[n]}$ denote its power set. For a family $\mathcal{F}\subset 2^{[n]}$, write

$$\delta(\mathcal{F})=\max\{|A\triangle B|:A,B\in\mathcal{F}\}$$

and define its diametral overflow by

$$\kappa_u(\mathcal{F})=\min\{|\mathcal{F}\setminus\mathcal{K}_A(n,u)|:A\subset[n]\},$$

where $\mathcal{K}_A(n,u)$ is a Kleitman extremal family of diameter $u$. Let $u$ be a positive integer, and write $u=2d$ or $u=2d+1$ according to its parity. Overflow conjecture. There exists an absolute constant $c$ such that, whenever $\delta(\mathcal{F})\leq u$ and $n>cu$,

$$\kappa_u(\mathcal{F})\leq\binom{n-2}{d-1}\quad\text{if }u=2d,$$

while

$$\kappa_u(\mathcal{F})\leq2\binom{n-3}{d-1}\quad\text{if }u=2d+1.$$

The bounds are motivated by the examples $\mathcal{B}(n,2d)$ and $\mathcal{G}(n,2d+1)$, for which the corresponding overflows attain these values; the conjecture asserts that these examples give universal upper bounds when $n$ is sufficiently larger than $u$.