The answer you link to contains this self-contradictory statement.
The minimum temperature required to fuse hydrogen is about 100 million Kelvin, which is about six times the temperature in the core of our Sun.
I cannot guess what the author means, but the statement is incorrect. The fusion of protons (the nuclei of hydrogen) occurs at about 15 million Kelvin in the Sun and at slightly lower temperatures in lower mass stars.
The energy generation rate by nuclear fusion isn't an on/off process, it occurs at a rate determined by the temperature and density of the reactants. However, the temperature dependence is strong - about $T^4$ in a star like the Sun - which means the fusion rate ramps up quickly with increasing temperature. Because of that, it is often said that there is a temperature threshold for [significant] fusion.
In the case of Jupiter, what is being discussed is not the fusion of protons but the fusion of deuterium nuclei (deuterons - consisting of a proton and a neutron). It turns out that this is far easier to accomplish and the "threshold temperature" for significant fusion is about 500 thousand Kelvin.
How close is Jupiter to that? Not very close. To a reasonable approximation, the central temperature of a ball of gas is proportional to $M/R$. The threshold temperature for deuterium burning is reached in objects that have about the same radius as Jupiter but about 13 times the mass. Using the proportionality above we might guess that the central temperature of Jupiter is at least an order of magnitude too low to initiate significant fusion of deuterium.
If you add mass to Jupiter and allow it to settle, then the core will become hotter and eventually, if enough mass is added, it would begin deuterium fusion, with almost exactly (chemical composition plays a minor role) the same temperature and interior pressure as a low-mass brown dwarf.