Asymptotically optimal time-and-budget conjecture for k-connectivity

Let $k$ be a positive integer, let $\varepsilon>0$, and let a $(t,b)$-strategy $B$ of Builder be a strategy that observes at most $t$ arriving edges and purchases at most $b$ of them. A graph is $k$-connected if it remains connected after deletion of any set of fewer than $k$ vertices. Asymptotically optimal time-and-budget conjecture for $k$-connectivity. If

$$t\ge(1+\varepsilon)n\log n/2\quad\text{and}\quad b\ge(1+\varepsilon)kn/2,$$

then there exists a $(t,b)$-strategy $B$ of Builder such that

$$\lim_{n\to\infty} \mathbb{P}(B_t\text{ is $k$-connected})=1.$$

The claim is stated as an open extension of the established construction for minimum degree: allowing asymptotically optimal time and budget should suffice to construct a $k$-connected graph. The source notes that the case $k=1$ is trivially true.