Is there a nice characterisation for the set of extensions of a given state?

Let $rhoinmathrm{D}(mathcal H)$ be a state in some (finite-dimensional) Hilbert space $mathcal H$ and suppose that $operatorname{rank}(rho)=r$.
This means that we can write it as
$$rho = sum_{k=1}^r p_k mathbb P_{u_k}, quadtext{with}quad p_k>0, ,,sum_k p_k=1, ,, mathbb P_{u_k}equiv lvert u_krangle!langle u_krvert.tag1$$

The set of purifications of $rho$ is easily characterised as the set of states $newcommand{ket}[1]{lvert #1rangle}ketpsiinmathcal Hotimes mathcal H_A$, with $dimmathcal H_A=r$, that have the form
$$ketpsi=sum_{k=1}^r sqrt{p_k} ket{u_k}otimesket{v_k},tag2$$
for any orthonormal basis ${ket{v_k}}_{k=1}^rsubset mathcal H_{A}$.
We can of course also consider purifications with larger ancillary spaces, but those would be trivially reducible to the case with $dimmathcal H_A=r$.

Consider now the more general set of extensions of $rho$.
This is the set of states $tilderhoinmathrm{D}(mathcal Hotimesmathcal H_A)$, for some auxiliary space $mathcal H_A$, such that $operatorname{Tr}_Atilderho=rho$. An example of a non-pure extension for a qubit can be found e.g. in this answer.

One way to characterise extensions is to observe that any extension can be written as the partial trace of a purification. As per our observation above about purifications, we can take a purification $ketpsi$ that uses a bipartite auxiliary space, $mathcal H_A=mathcal H_Botimesmathcal H_C$, so that $ket{v_k}inmathcal H_Botimesmathcal H_C$, and then tracing out $mathcal H_C$ gives
begin{align}newcommand{ketbra}[2]{lvert #1rangle!langle #2rvert}
tilderho
= sum_k sqrt{p_j p_k} ketbra{u_j}{u_k}otimes operatorname{Tr}_C[ketbra{v_j}{v_k}]
= sum_k sqrt{p_j p_k} ketbra{u_j}{u_k}otimes sigma_{jk},
tag3
end{align}
where $sigma_{jk}equiv operatorname{Tr}_C[ketbra{v_j}{v_k}]inmathrm{Lin}(mathcal H_B)$ is such that $operatorname{Tr}(sigma_{jk})=delta_{jk}$.
Any extension can be written as (3), as if $tilderho$ is a generic extension, then its purification has the form in Eq. (2), and then partial tracing we get the form in Eq. (3).

Are there nicer/more elegant/terser characterisations for the set of extensions of $rho$?