'Gay cake' row in Supreme Court
The 2015 ruling against Ashers bakery has already been challenged in every other available court.
The Christian owners of a bakery found to have discriminated in refusing to make a "gay cake'' have said the law "extinguishes'' their consciences.
A lower court ruled the decision of family-run Ashers not to bake the product iced with the slogan Support Gay Marriage in 2014 was discriminatory after a legal challenge supported by Northern Ireland's Equality Commission.
The order was placed at its Belfast shop by gay rights activist Gareth Lee.
The UK's highest court, sitting in Belfast, is to hear legal arguments surrounding the case on Tuesday.
Daniel McArthur, general manager of Ashers, said: "The Equality Commission has pushed for an interpretation of the law which extinguishes our conscience.
"They think that some people are more equal than others.''
Mr McArthur said he should enjoy a basic right to live by his beliefs.
"But some people want the law to make us support something with which we disagree.''
He said people across society were worried about what this could mean.
The case against Ashers was taken by Mr Lee with support from Northern Ireland's Equality Commission.
Controversy first flared when Mr Lee, a member of the LGBT advocacy group Queer Space, ordered a cake featuring Sesame Street puppets Bert and Ernie for a private function marking International Day Against Homophobia.
He placed the order in person at Ashers' Belfast city centre branch in May 2014.
It was accepted and he paid in full but, two days later, the company called to say it could not proceed due to the message requested.
In the original court case, District Judge Isobel Brownlie ruled that religious beliefs could not dictate the law and ordered the firm to pay damages of £500.
Mounting an appeal, Ashers contended that it never had an issue with Mr Lee's sexuality, rather the message he was seeking to put on the cake.
The business said the slogan was inconsistent with its deeply held religious beliefs.