diff --git a/image_analysis/bragg_integration/BraggIntegrationEngine.h b/image_analysis/bragg_integration/BraggIntegrationEngine.h
index 4b4430808..fc0020638 100644
--- a/image_analysis/bragg_integration/BraggIntegrationEngine.h
+++ b/image_analysis/bragg_integration/BraggIntegrationEngine.h
@@ -87,6 +87,10 @@ constexpr float MAX_STENCIL_GROW_OVER_R3 = 2.0f;
 // summation (box-sum) intensity when the profile result disagrees with the summation seed by more than
 // this many box-sum sigmas (a real fit agrees within counting noise, so the margin is generous).
 constexpr double PROFILE_SUMMATION_MAX_NSIGMA = 10.0;
+// The runaway is a weak reflection's. On a strong spot the profile reads a few per cent above the box
+// sum (the box misses the tails), which at high I/sigma is many box sigmas - a real fit, not a
+// runaway. So a profile above a box sum of more than this many sigmas is kept.
+constexpr double PROFILE_SUMMATION_STRONG_NSIGMA = 20.0;
 
 // MINPK keeps a reflection while enough of its expected profile is readable. It says nothing about
 // WHERE the unreadable part is, and the two are not the same question. Renormalising over the pixels
diff --git a/image_analysis/bragg_integration/BraggIntegrationEngineCPU.cpp b/image_analysis/bragg_integration/BraggIntegrationEngineCPU.cpp
index 0d283d7a1..28c4e52ed 100644
--- a/image_analysis/bragg_integration/BraggIntegrationEngineCPU.cpp
+++ b/image_analysis/bragg_integration/BraggIntegrationEngineCPU.cpp
@@ -890,7 +890,9 @@ std::vector<Reflection> BraggIntegrationEngineCPU::ProfilePasses(const Sampler &
         // both - excluded to a neighbour, or unreadable - the fit's full-profile intensity has to be
         // scaled down to that same disk before the two are comparable. Nothing dropped gives 1.
         const double guard_scale = fo.m_all > 0.0f ? fo.m_read / fo.m_all : 1.0;
-        if (std::abs(I * guard_scale - rh.I) > PROFILE_SUMMATION_MAX_NSIGMA * rh.sigma) {
+        const double guard_diff = I * guard_scale - rh.I;
+        const bool strong_above = guard_diff > 0 && rh.I > PROFILE_SUMMATION_STRONG_NSIGMA * rh.sigma;
+        if (std::abs(guard_diff) > PROFILE_SUMMATION_MAX_NSIGMA * rh.sigma && !strong_above) {
             I = rh.I;
             sigma = rh.sigma;
             var_bkg = rh.var_bkg;
diff --git a/image_analysis/bragg_integration/BraggIntegrationEngineGPU.cu b/image_analysis/bragg_integration/BraggIntegrationEngineGPU.cu
index 34e242fab..43ac653f3 100644
--- a/image_analysis/bragg_integration/BraggIntegrationEngineGPU.cu
+++ b/image_analysis/bragg_integration/BraggIntegrationEngineGPU.cu
@@ -873,7 +873,10 @@ __global__ void fit(const int32_t *img, const uint32_t *owner, const float *px_x
             // neighbour, or unreadable - scale the full-profile intensity back to the disk the seed
             // read; nothing dropped gives exactly 1.
             const float gs = s_mall > 0.0f ? s_mread / s_mall : 1.0f;
-            if (fabsf(I * gs - I_seed[i]) > (float) PROFILE_SUMMATION_MAX_NSIGMA * sigma_seed[i]) {
+            const float diff = I * gs - I_seed[i];
+            const bool strong_above = diff > 0.0f
+                    && I_seed[i] > (float) PROFILE_SUMMATION_STRONG_NSIGMA * sigma_seed[i];
+            if (fabsf(diff) > (float) PROFILE_SUMMATION_MAX_NSIGMA * sigma_seed[i] && !strong_above) {
                 I = I_seed[i];
                 sigma = sigma_seed[i];
                 var_bkg = varbkg_seed[i];