Let $T^*$ denote upper triangular matrices (of the appropriate size) with positive diagonal entries and $\mathrm{UT}$ upper triangular matrices with all diagonal entries equal to 1.

Does every (abstract group) embedding $\varphi:\mathrm{UT}(n,\mathbb{R})\to\mathrm{UT}(m,\mathbb{R})$ extend to $\bar{\varphi}:T^*(n,\mathbb{R})\to T^*(m,\mathbb{R})$?

(In the other direction, any $\bar{\varphi}:T^*(n,\mathbb{R})\to T^*(m,\mathbb{R})$ restricts to a homomorphism $\mathrm{UT}(n,\mathbb{R})\to\mathrm{UT}(m,\mathbb{R})$, since $\mathrm{UT}$ is the derived subgroup of $T^*$.)

An affirmative answer to this question implies a relatively easy affirmative answer to this question. I've recently answered the latter independently, but wonder if there's a general result that could be used, and which I should know about. I asked the current question on Math.stackexchange but without success.

EDIT: Florian Eisele has shown that as stated above the answer to the original question is no. This makes me wonder if there's a reasonably natural reformulation for which the answer is yes. For the sake of asking a concrete question, let me hazard the following.

Does every embedding $\varphi:\mathrm{UT}(n,\mathbb{Q})\to\mathrm{UT}(m,\mathbb{Q})$ extend to $\bar{\varphi}:T^*(n,\mathbb{Q})\to T^*(m,\mathbb{Q})$?