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EDIT: I've now posted this question on MathOverflow.

Here $T^*$ means upper triangular with positive diagonal entries and $\mathrm{UT}$ means upper triangular matrices with all diagonal entries equal to 1. For what it's worth, the $\varphi=\varphi_n$ that I have in mind give rise to a free and transitive (aka simply transitive) action of the domain on $\mathbb{R}^{m-1}$ where $m=\frac{n(n-1)}{2}+1$.

Does every injective group homomorphism $\varphi:\mathrm{UT}(n,\mathbb{R})\to\mathrm{UT}(m,\mathbb{R})$ admit an extension $\bar{\varphi}:T^*(n,\mathbb{R})\to T^*(m,\mathbb{R})$?

An affirmative answer to this question implies an affirmative answer to this question.

(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 $T^*$.)

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