A topic from AEG's L5R CCG Rules forum, copied for Oracle of the Void from a copy of the forum saved in October 2015, so pages it didn't save are missing. Players' names are left out; the rules officials are named. Rulings are as they stood on these dates; later rulings, the rulebooks and the Rules Archive may differ.
A player, 5 March 2009
If my opponent, who controls Kuro Sensei, has a dishonored personality and Kakita Kaisei issues and unrefusable challenge to one of his personalities and I use Kaisei's reaction targeting the dishonored personality, can the challenge be refused (by Kuro Sensei's text) or does Kaisei's ability make my opponent unable to pay the additional cost of refusing thus making my opponent's personality unable to refuse the challenge?
Relevant Texts:
Kakita Kaisei Crane Clan Samurai • Courtier • Duelist • Magistrate Reaction: After Kaisei issues a challenge: Target a Personality controlled by the challenged player. Refusing the challenge has the additional cost of Dishonoring the target. (If it already has the cost of Dishonoring that target, the challenge may not be refused.)
Kuro Sensei Lion. Mantis. Monk. Phoenix Toturi's Army Sensei Your personalities may refuse challenges,[sic] that are normally unrefusable. Your personalities will not issue challenges. If one of your personalities refuses a challenge with no given penalty for refusal, you lose 3 honor and the personality becomes dishonored.
Brook Cunningham (rules official), 5 March 2009
If my opponent, who controls Kuro Sensei, has a dishonored personality and Kakita Kaisei issues and unrefusable challenge to one of his personalities and I use Kaisei's reaction targeting the dishonored personality, can the challenge be refused (by Kuro Sensei's text) or does Kaisei's ability make my opponent unable to pay the additional cost of refusing thus making my opponent's personality unable to refuse the challenge?
In general, Kuro Sensei will not waive a cost of refusal. Which means, in your specific example, an already dishonorable Personality will not be able to pay the cost of refusing Kaisei's challenge.
"Normally unrefusable" challenges (per Kuro Sensei's text) are ones that are either explicitly unrefusable in the case of pre-SE-legal duels, or ones that do not explicitly say they may be refused in the case of SE-legal and later duels. Kuro Sensei will not analyze the situation at hand to see if you can refuse or not, and then allow you to do so regardless.