The three conjectural sporadics in the entropic order on a parabolic quotient of $S_6$

In the above notation, let the displayed arrays denote the corresponding elements of the parabolic quotient, and let $\rhd$ denote the entropic partial order.

Three sporadic entropic-order conjectures. The following three relations hold:

$$\begin{array}{rcl} \mathbf{37}\rhd\mathbf{11}:\quad \left(\begin{array}{ccc}a & c & f\\b & d & e\end{array}\right)&\rhd&\left(\begin{array}{ccc}a & b & d\\f & c & e\end{array}\right),\\ \mathbf{43}\rhd\mathbf{11}:\quad \left(\begin{array}{ccc}a & d & e\\b & c & f\end{array}\right)&\rhd&\left(\begin{array}{ccc}a & b & d\\f & c & e\end{array}\right),\\ \mathbf{49}\rhd\mathbf{11}:\quad \left(\begin{array}{ccc}a & d & f\\b & c & e\end{array}\right)&\rhd&\left(\begin{array}{ccc}a & b & d\\f & c & e\end{array}\right). \end{array}$$

These are the remaining sporadic relations identified in the paper; the preceding discussion explains that the author was unable to prove them and that their structure does not yield to the techniques used for the other relations.