Bosonic Fradkin-Tseytlin equations unfolded. Irreducible case

We factorize 4d Fradkin-Linetsky higher spin conformal algebra by maximal ideal $I^1-\alpha$ and construct irreducible infinite-dimensional modules $M_\alpha$ of 4d conformal algebra that are parameterized by real number $\alpha$. It is shown that independently of $\alpha$ unfolded system of equations corresponding to each $M_\alpha$ describes collection of Fradkin-Tseytlin equations for all spins $s=1,\dots,\infty$ with zero multiplicity.