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A Latin square \((S, \cdot)\) is said to be \((3, 1, 2)\)-\emph{conjugate-orthogonal} if \(x \cdot y = z \cdot w\), \(x \cdot_{312} y = z \cdot_{312} w\) imply \(x = z\) and \(y = w\), for all \(x, y, z, w \in S\), where \(x_3 \cdot_{312} x_1 = x_2\) if and only if \(x_1 \cdot x_2 = x_3\).
Such a Latin square is said to be \emph{holy} \(((3,1,2)\)-HCOLS for short) if it has disjoint and spanning holes corresponding to missing sub-Latin squares.
Let \((3,1,2)\)-HCOLS\((h^n)\) denote a \((3,1,2)\)-HCOLS of order \(hn\) with \(n\) holes of equal size \(h\).
We show that, for any \(h \geq 1\), a \((3,1,2)\)-HCOLS\((h^n)\) exists if and only if \(n \geq 4\), except \((n,h) = (6,1)\), and except possibly \((n,h) = (10,1)\) and \((4,2t+1)\) for \(t \geq 1\).
Let \((3,1,2)\)-ICOILS\((v,k)\) denote an idempotent \((3,1,2)\)-COLS of order \(v\) with a hole of size \(k\).
We prove that a \((3,1,2)\)-ICOILS\((v,k)\) exists for all \(v \geq 3k+1\) and \(1 \leq k \leq 5\), except possibly \(k = 4\) and \(v \in \{35, 38\}\).