The Ostaszewski square, and homogeneous Souslin trees

Abstract: Assume GCH and let $\lambda$ denote an uncountable cardinal.
We prove that if $\square_\lambda$ holds, then this may be  witnessed by a coherent sequence $\left\langle C_\alpha \mid \alpha<\lambda^+\right\rangle$ with the following remarkable guessing property:

For every sequence $\langle A_i\mid i<\lambda\rangle$ of unbounded subsets of $\lambda^+$, and every limit $\theta<\lambda$, there exists some $\alpha<\lambda^+$ such that $\text{otp}(C_\alpha)=\theta$, and the $(i+1)_{th}$-element of $C_\alpha$ is a member of $A_i$,  for all $i<\theta$.

As an application, we introduce the first construction of an homogeneous Souslin tree at the successor of a singular cardinal.
In addition, as a by-product, a theorem of Farah and Velickovic (see [FV]), and a theorem of Abraham, Shelah and Solovay (see [AShS:221]) are generalized to cover the case of successors of regulars

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