Call a direct structure reflecting when every morphism between objects of equal degree is a split epimorphism. In a reflecting direct category, every non-invertible-in-degree morphism strictly raises degree, and the strict downset is as large as a proper sieve can be:

If the direct structure is reflecting, then every proper sieve $𝑆$ on $𝑥$ refines into the strict downset:

Suppose $𝑓\in 𝑆$ with $𝑓:𝑦\to 𝑥$ of equal degree. By reflection $𝑓$ has a section $𝑠$, and closure under precomposition gives $𝑠\star 𝑓={\text{𝗂𝖽}}_{𝑥}\in 𝑆$, contradicting properness.

So every morphism in $𝑆$ strictly raises degree. That is, every morphism in $𝑆$ is also a member of $\text{𝖲𝗍𝗋𝗂𝖼𝗍𝖣𝗈𝗐𝗇}(𝑥)$.