When Draper regained his place as world number four following his run to the Queen's semi-finals, it was a significant moment.
As fourth seed, Draper was guaranteed to avoid world number one Jannik Sinner or two-time defending champion Carlos Alcaraz – the heavy favourites for the title - until at least the semi-finals.
Draper, however, was quick to urge caution about placing too much importance on the seeding until the draw was made - and he has been proven right.
If he beats Baez, Draper could face 2017 finalist Marin Cilic and 28th seed Alexander Bublik - who beat the Briton at the French Open and has since won the Halle title - in the second and third rounds.
Czech 15th seed Jakub Mensik, a huge server who could be a serious threat on grass, is a prospective fourth-round opponent before the looming spectre of sixth seed Djokovic or Australian 11th seed Alex de Minaur in the last eight.
British number two Jacob Fearnley has been drawn against much-hyped Brazilian teenager Joao Fonseca, while 2022 semi-finalist Cameron Norrie faces Spanish veteran Roberto Bautista Agut, who also reached the last four here in 2019.
Tarvet, 21, starts against another qualifier in Switzerland's Leandro Riedi.