A negative answer to a question on tilting objects and two-term complexes

Let $M$ be a silting object in an idempotent complete algebraic triangulated category $\mathcal{T}$. Put $B={\mathrm{E}\mathrm{n}\mathrm{d}}_{\mathcal{T}}(M)$, and let ${\mathbb{P}}_M\text{\hspace{0.17em}⁣}:\mathrm{p}\mathrm{r}(M)\to K^{[-1,0]}(\mathrm{p}\mathrm{r}\mathrm{o}\mathrm{j}B)$ be the presentation functor associated with $M$. It was recently asked whether ${\mathbb{P}}_M(T)$ must be tilting whenever $T\in \mathrm{p}\mathrm{r}(M)$ is a tilting object. We answer this question in the negative by giving an explicit finite-dimensional example. Namely, for $$\mathrm{\Lambda }=k(1\stackrel{\alpha }{\to }2\stackrel{\beta }{\to }3\stackrel{\gamma }{\to }4)\mathrm{/}(\alpha \beta \gamma ),$$ we construct a silting object $M\in K^b(\mathrm{p}\mathrm{r}\mathrm{o}\mathrm{j}\mathrm{\Lambda })$ and a tilting object $T=\mathrm{\Sigma }\mathrm{\Lambda }\in \mathrm{p}\mathrm{r}(M)$ for which $${\mathrm{d}\mathrm{i}\mathrm{m}}_k{\mathrm{H}\mathrm{o}\mathrm{m}}_{K^b(\mathrm{p}\mathrm{r}\mathrm{o}\mathrm{j}B)}({\mathbb{P}}_M(T),{\mathrm{\Sigma }}^{-1}{\mathbb{P}}_M(T))=1.$$ Thus ${\mathbb{P}}_M(T)$ is a two-term silting complex but not a tilting complex.