What is the status of the assertion "There are arbitrarily large cardinals with the tree property"?

Let me address your question in the comments.

If one starts with a weakly compact cardinal $\kappa,$ then there exists a forcing extension in which all cardinals and cofinalities are preserved (in particular, $\kappa$ remains a regular limit cardinal) and in which the tree property holds at $\kappa$. In fact $\kappa$ can become the least weakly Mahlo cardinal.

This result is due to Boos, see Boolean extensions which efface the Mahlo property.

As far as I know the question of whether the least weakly inaccessible cardinal can have the tree property is open (it is asked in Boos paper).

Paul B. Larson mentions a partial result in A Brief History of Determinacy, page 48. Sadly he doesn't give a precise source.

Foreman, Magidor and Schindler showed that if there exist infinitely many cardinals $δ$ above the continuum such that the tree property holds at $δ$ and at $δ^+$, then [the axiom of projective determinacy] holds. The hypothesis of this statement had been shown consistent relative to the existence of infinitely many supercompact cardinals by James Cummings and Foreman. It is not known whether the conclusion can be strengthened to $\text{AD}^{L(\mathbb{R})}$.

So the tree property for all successors >$\omega_1$ has to be a very strong statement, at least as strong as the existence of infinitely many Woodin cardinals. This is the best partial/related result I know of.

EDIT: See this paper, which has the following abstract:

Starting from a strong cardinal and a measurable cardinal above it, we construct a model of ZFC, in which, for every singular cardinal $δ$, $δ$ is strong limit, $2^δ = δ^{+++}$, and the tree property holds at $δ^{++}$. It answers a question of Friedman, Honzik and Stejskalova. We also produce, relative to the existence of a strong cardinal and two measurable cardinals above it, a model of ZFC in which the tree property holds at all regular even cardinals. The result answers questions of Friedman, Halilovic and Honzik .

Which answers the question from the title, i.e. it is possible to have arbitrarily large cardinals with the tree property (relatively to very large cardinals).