Here's an unfolded ASCII-art version of our cube. The longest path has length 37.

*********************
* 3  |    # 8  |    * 
*    |    #    |    * 
* ---+--- # ---+--- * 
*    |    #    |    * 
*    |    #    |    *   
*###################* 
* 7  |    # 6  |    * 
* __/     #     \__ * 
*         #         * 
*         #         * 
*         #         * 
*********************************************************************************
* 7  |    # 6  |    * 6  |    # 8  |    * 8  |    # 3  |    * 3  |    # 7  |    * 
*    |    #    |    * __/     #    |    *    |    #    |    * __/     #    |    * 
* ---+--- # ---+--- *      __ # ---+--- * ---+--- # ---+--- *      __ # ---+--- * 
*    |    #    |    *     /   #    |    *    |    #    |    *     /   #    |    *   
*    |    #    |    *    |    #    |    *    |    #    |    *    |    #    |    *   
*###################*###################*###################*###################* 
* 5  |    # 4  |    * 4  |    # 2  |    * 2  |    # 1  |    * 1  |    # 5  |    * 
*     \__ # __/     * __/     #     \__ *    |    #    |    *    |    #    |    * 
* __      #      __ *         #         * ---+--- # ---+--- * ---+--- # ---+--- * 
*   \     #     /   *         #         *    |    #    |    *    |    #    |    * 
*    |    #    |    *         #         *    |    #    |    *    |    #    |    * 
*********************************************************************************
* 5  |    # 4  |    *
*    |    # __/     *
* ---+--- #         *
*    |    #         *
*    |    #         *
*###################*
* 1  |    # 2       *
*     \__ #         *
* __      # __      *
*   \     #   \     *
*    |    #    |    *
*********************