Using notation by the [answer](https://mathoverflow.net/a/460118) by Hamkins, I'll prove that $\mathbb c_{\mathbb R}\ge\min\{\mathfrak s,\mathfrak b\}$.

Thanks to Theorem 8.11 of Halbeisen's book [Combinatorial Set Theory](https://doi.org/10.1007/978-1-4471-2173-2), we know that $\min\{\mathfrak s,\mathfrak b\}$ is equal to the smallest cardinality of any family $\mathcal P$ of maps from $[\omega]^2$ to $2$ such that there is no infinite subset $H\subseteq\omega$ that is almost homogeneous for all maps in $\mathcal P$. Let us fix $\lambda<\min\{\mathfrak s,\mathfrak b\}$.

Assume we have some bounded sequences $\langle x_n^\alpha: n<\omega\rangle$ with values in $\mathbb R$ for all $\alpha<\lambda$. We want to prove that we can make all of them converge with the same subindex set. For all $\alpha<\lambda$, let's define $f_\alpha(\{n,m\}):=0$ if $n<m$ and $x_n^\alpha<x_m^\alpha$ and $f_\alpha(\{n,m\}):=1$ if $n<m$ and $x_n^\alpha\ge x_m^\alpha$.

Then the family $\mathcal P:=\{f_\alpha: \alpha<\lambda\}$ has cardinality $\lambda$, hence there is an infinite subset $H\subseteq\omega$ that is almost homogeneous for all maps in $\mathcal P$. Let $\sigma:\omega\to H$ be the (only) order-preserving map.

We want to prove that $\langle x^\alpha_{\sigma(k)}:k<\omega\rangle$ converges for all $\alpha<\lambda$. So fix some $\alpha<\lambda$.

If $f_\alpha(\{n,m\})=0$ for almost all $\{n,m\}\in [H]^2$, then this means that for almost all $a<b<\omega$ we have $x^\alpha_{\sigma(a)}<x^\alpha_{\sigma(b)}$ and, since the sequence $\langle x^\alpha_n:n<\omega\rangle$ was bounded, the subsequence $\langle x^\alpha_{\sigma(k)}:k<\omega\rangle$ is a bounded increasing sequence, so it converges. If instead $f_\alpha(\{n,m\})=1$ for almost all $\{n,m\}\in [H]^2$, then this means that for almost all $a<b<\omega$ we have $x^\alpha_{\sigma(a)}\ge x^\alpha_{\sigma(b)}$ and, since $\langle x^\alpha_n:n<\omega\rangle$ is bounded, the subsequence $\langle x^\alpha_{\sigma(k)}:k<\omega\rangle$ is a bounded nondecreasing sequence, so it converges.

So, this used some specific properties of $\mathbb R$, like the ordering, but I guess it could be generalized to Banach spaces or, as I said in the [comments](https://mathoverflow.net/questions/460114/extracting-a-common-convergent-indexing-from-an-uncountable-family-of-sequences#comment1192263_460118) to Hamkins's [answer](https://mathoverflow.net/a/460118), even to generic convergence spaces. The question whether $\mathbb c=\mathbb c_{\mathbb R}$ is still open and interesting. Another interesting fact would be trying to prove $\mathbb c_{\mathbb R}=\min\{\mathfrak s,\mathfrak b\}$, that would be equivalent to $\mathfrak b\ge\mathbb c_{\mathbb R}$.

I would love to write more conclusions but where I am it is already past 2 a.m. so I should rest. But probably tomorrow or in the next few days I'll try looking more into detail.