(responding to @dgesang here, because of max comment depth)
The conventionally-grown food looks similar, tastes similar, and people have been eating it for decades with no major, obvious consequences. So at first glance they are equivalent. So it seems reasonable that the burden would be on proving a difference, rather than an equivalence.
The dietary effects on health are poorly understood. Nutrition studies are fraught with difficulties and have spawned a resurgence in methodology concerning "measurement error" in the field of statistics. You don't really know how much pizza you ate last year, but the survey will ask you to estimate it somehow... and this kind of measurement is 'noisy'.
Conventionally-grown food looks like organically grown food on steroids, to me. I suppose from the outside a human on steroid may look healthy -- Mr Universe, even -- but we know all is not well inside.
I'm not saying that's really related, but there is substantial evidence for real differences in the makeup of food grown conventionally vs organically.
Citations 10-15 in the paper the original article pertain to this - protein expression differences across fertilization types.
"... The abundance of 111 protein spots varied significantly between fertilisation regimes. Flag leaf N and P composition were significant drivers of differences in protein spot abundance, including major proteins involved in nitrogen remobilisation, photosynthesis, metabolism and stress response."
And organically-grown food have been eating it for EONS with no major, obvious consequences. 'Conventionally-grown food' have just been thrown on the market a few decades ago without knowing any long-term effects to humans eating them or to the environment they are grown in. I don't care about proof for equivalence or difference, I only care about safety. And any food should be proven to be safe BEFORE being released, not decades later.
The conventionally-grown food looks similar, tastes similar, and people have been eating it for decades with no major, obvious consequences. So at first glance they are equivalent. So it seems reasonable that the burden would be on proving a difference, rather than an equivalence.