Current proposed technology, specifically initiatives like Breakthrough Starshot, relies on laser-driven photonic propulsion to accelerate lightweight lightsails to approximately one-fifth to a quarter of the speed of light.
The claim states that current technology allows accelerating space probes to a quarter of the speed of light. Papers [3] and [10] discuss the Breakthrough Starshot initiative, which is actively designing and testing photonic propulsion systems using laser beams to accelerate lightsails to relativistic speeds around 0.2c (one-fifth of the speed of light, very close to a quarter). Since these papers support the feasibility and ongoing development of technology aiming for speeds on this order of magnitude, and no papers refute the proposition, the verdict is SUPPORTED.