Would anyone know an example of a simplicial set having the "spine-lifting" property while not being a quasicategory? Definition:

*

*By a simplicial set we mean a Set-valued presheaf on the category $\mathbf{\Delta}$ of nonempty finite totally-ordered-sets with non-strict order-preserving functions between them. The category of simplicial sets is denoted $\mathrm{sSet}$.
For integer $k\geq0$ let $\Delta^k$ denote the simplicial set represented as a presheaf by $[k] = (\{0,1,2,\ldots,k\},\leq)$; we call this the (simplicial set model of the) $k$-simplex.

*For integer $k\geq0$ let $L_k \subset \Delta^k$ denote the spine of the $k$-simplex.

*Let us say a given simplicial set $X$ has property $\mathscr{P}$, the "spine-lifting property",
if for every integer $k\geq0$ the restriction function $\mathrm{Hom}_{\text{sSet}}(\Delta^k , X) \rightarrow \mathrm{Hom}_{\text{sSet}}(L_k , X)$ is surjective.

My question is, can there be a simplicial set $X$ such that $X$ has property $\mathscr{P}$ but $X$ is not a quasicategory? I suspect the answer is "yes", otherwise we would just define a quasicategory as a simplicial set having property $\mathscr{P}$.
Note: according to the comment on this question, every quasicategory has property $\mathscr{P}$. I would like to know if the converse has a counterexample.
 A: One source of examples is simplicially enriched categories not all of whose mapping simplicial sets are Kan complexes (so non-fibrant ones). The coherent nerve of a simplicially enriched category always has spine fillers, which you can prove as follows. By adjunction, to produce an extension of $L_n\to NC$ to $\Delta^n$ is to produce an extension of $\mathfrak C(L_n)\to C$ to $\mathfrak C(\Delta^n)$. But the map  $\mathfrak C(L_n)\hookrightarrow \mathfrak C(\Delta^n)$ has a left inverse because the domain has constant mapping simplicial sets and the same objects as the codomain, so a lift is given by composition with this left inverse. I learned this in Markus Land's book on infinity-categories.
So, it's sufficient to find a simplicially enriched category whose coherent nerve is not a quasicategory. Such an object necessarily will have non-Kan mapping simplicial sets. Perhaps a melodramatic example is the simplicially enriched category of small quasicategories (for instance with a cardinality cutoff in order to make it small). I believe $\mathfrak C(\Delta^3)$ should also work, since $\operatorname{Map}(0,3)\cong \Delta^1\times\Delta^1$, and there are unfillable inner 2-horns corresponding to trying to invert one of the sides of the square.
