See the argument of "Homotopy of posets" by G. Raptis, Homology Homotopy Appl. 12 (2010), no. 2, 211–230, Remark 4.7. It would suffice to prove that for any cardinal $\mu$, there exists a cardinal $\nu > \mu$ such that the topological space $\mathcal{T}(P_\nu)$ is homotopy equivalent to a CW-complex.

Your setting is very closed to the setting of model categories with a prescribed class of fibrant objects like in tac.mta.ca/tac/volumes/29/23/29-23.pdf because your pseudo-generating set of trivial cofibrations determines the fibrant objects (you probably already know the reference but in case you don't, I give it).

In Cisinski's homotopy theory of toposes, the minimal model structure is always right proper (Remark 4.9 of his paper). And I don't believe that Condition 3 holds (I am not sure actually, it's why I am writing this comment).

Nobody mentioned here that Vopenka's principle has interesting consequences in homotopy theory. A good starting point is the nLab: ncatlab.org/nlab/show/Vop%C4%9Bnka%27s+principle

@JacquesCarette If the natural numbers are embedded in the whole set of integers, is the theorem "$2$ is a nonnegative integer" a junk theorem or not ?

What is supposed to be the suspension of a category ?

@MikhailBondarko A category does not have to be locally small. The only precaution you have to take is not asserting that a proper class is a set if it is not a set.

There is an interesting discussion about the two out of six property in "Homotopy Limit Functors on Model Categories and Homotopical Categories" by Dwyer, Hischhorn, Kan and Smith (Mathematical Surveys and Monographs 113 American Mathematical Society).

Concerning the interest of weak Hausdorff spaces, see my answer http://mathoverflow.net/a/204627/24563.

Yes I made this suggestion because $S\mapsto hom(-,S)$ commutes with products.

If $C(S)$ is your colimit and $L(S)$ your limit, did you try to calculate $C(\varnothing)$, $L(\varnothing)$, $C(S \times T)$, $L(S \times T)$, $C(\{0\})$, $L(\{0\})$ ?

Note that your set can be a proper class. Using Vopenka's principle, in the category of simplicial sets, there is one left determined model structure with respect to $\Delta[0]\to K$ with $K$ nonempty.

I was not expecting any answer to my soft question about a "conjecture" which is not even a conjecture :-). So I'll mark your post as the answer.

Could you say please what proof assistant you are using ?