Human rights group Amnesty International is campaigning on his behalf and his plight was highlighted this week in the United States Congress.
Mr Genser said his client was one of Sudan's "Lost Boys", displaced by the country's civil war.
He resettled in the United States, studying at La Salle University in Philadelphia and Harvard University, before moving to Cambridge University.
Returning to his home country on 28 July to hold a youth forum, he was arrested and taken directly to custody.
Mr Genser said his client had called for the country's current leaders to step down so that younger people could take over and achieve peace.
"This has become a real problem for the government in South Sudan, which then decides to target him for arrest and arbitrary detention because he was being a very effective critic," he said.
On Thursday, Congresswoman Madeleine Dean, who "knew Peter as a brilliant student and leader" while teaching at La Salle University, drew attention to his detention in the US House of Representatives.
Mr Genser said charges being considered by the South Sudanese authorities included treason and terrorism, both of which carry the death penalty.
"Somebody like him needs to be on the front lines fighting for freedom, democracy and human rights - not arbitrarily detained in a modern-day hellhole in clear violation of his rights under international law and for crimes he did not commit," said Mr Genser.
South Sudan fact file
The East African nation is world's youngest, having gained independence in 2011, but has been troubled by internal conflict for much of its existence
A 2017-18 Amnesty International report found approximately one third of the population had been displaced by the conflict
By the end of 2017, UNICEF estimated that more than 19,000 child soldiers had been used in the country's civil war
A Cambridge University spokeswoman said: "The university remains deeply concerned about Peter's welfare and his access to legal representation and the violation of his rights in accordance with the constitution of South Sudan, which guarantees all South Sudanese people liberty and security of person, due process, and freedom of expression and association."
Seif Magango, Amnesty International's deputy regional director for East Africa, said Mr Ajak's ongoing detention without charge was "absurd" and in breach of South Sudan's own constitution and international law.
"South Sudanese authorities must either release him so he can re-join his wife and children who miss him dearly, or charge him with an offence recognised under international law," he said